Question
If $\left[\begin{array}{cc}-1 & 0 \\ 0 & 1\end{array}\right]\left[\begin{array}{ll}a & b \\ c & d\end{array}\right]=\left[\begin{array}{cc}1 & 0 \\ 0 & -1\end{array}\right]$ find $a, b, c$ and $d$
✓
Answer
Given
$
\begin{aligned}
& {\left[\begin{array}{ll}
-1 & 0 \\
0 & 1
\end{array}\right]\left[\begin{array}{ll}
a & b \\
c & d
\end{array}\right]=\left[\begin{array}{cc}
1 & 0 \\
0 & -1
\end{array}\right]} \\
& \Rightarrow\left[\begin{array}{ll}
a+0 & -b+0 \\
0+c & 0+d
\end{array}\right]=\left[\begin{array}{cc}
1 & 0 \\
0 & -1
\end{array}\right] \\
& \Rightarrow\left[\begin{array}{cc}
-a & -b \\
c & d
\end{array}\right]=\left[\begin{array}{cc}
1 & 0 \\
0 & -1
\end{array}\right]
\end{aligned}
$
Comparing the corresponding elements
$
\begin{aligned}
& -a=1 \\
& \Rightarrow a=-1 \\
& -b=0 \\
& \Rightarrow b=0 \\
& c=0 \text { and } d=-1
\end{aligned}
$
Hence $a=-1, b=0, c=0, d=-1$.
$
\begin{aligned}
& {\left[\begin{array}{ll}
-1 & 0 \\
0 & 1
\end{array}\right]\left[\begin{array}{ll}
a & b \\
c & d
\end{array}\right]=\left[\begin{array}{cc}
1 & 0 \\
0 & -1
\end{array}\right]} \\
& \Rightarrow\left[\begin{array}{ll}
a+0 & -b+0 \\
0+c & 0+d
\end{array}\right]=\left[\begin{array}{cc}
1 & 0 \\
0 & -1
\end{array}\right] \\
& \Rightarrow\left[\begin{array}{cc}
-a & -b \\
c & d
\end{array}\right]=\left[\begin{array}{cc}
1 & 0 \\
0 & -1
\end{array}\right]
\end{aligned}
$
Comparing the corresponding elements
$
\begin{aligned}
& -a=1 \\
& \Rightarrow a=-1 \\
& -b=0 \\
& \Rightarrow b=0 \\
& c=0 \text { and } d=-1
\end{aligned}
$
Hence $a=-1, b=0, c=0, d=-1$.
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